On locally constructible spheres and balls
Please always quote using this URN:urn:nbn:de:0296-matheon-6283
- Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are: (1.) Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) (2.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.) (3.) All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.) (4.) Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)
Author: | Bruno Benedetti, Günter M. Ziegler |
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URN: | urn:nbn:de:0296-matheon-6283 |
Referee: | Günter M. Ziegler |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2009/07/21 |
Release Date: | 2009/07/21 |
Institute: | Technische Universität Berlin |
Zuse Institute Berlin (ZIB) | |
MSC-Classification: | 05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Axx Enumerative combinatorics For enumeration in graph theory, see 05C30 / 05A16 Asymptotic enumeration |
52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B22 Shellability | |
57-XX MANIFOLDS AND CELL COMPLEXES (For complex manifolds, see 32Qxx) / 57Mxx Low-dimensional topology / 57M25 Knots and links in S3 (For higher dimensions, see 57Q45) | |
Preprint Number: | 629 |